I prefer the A:EMH theme. I never watched the Justice League cartoon as a child and though it's certainly more iconic, my having not watched the show makes its iconicism lost on me. In a vacuum it sounds like generic "epic" music that I wouldn't necessarily associate with the Justice League unless I knew it was the Justice League theme.
Whereas the Avengers theme succeeds in being an Avengers song without sounding forced - they haven't written twenty-four lines of awkwardly put together lyrics explaining the powers of each superhero, it's just a simple hook over and over again that would have got me fucking hype as a kid. It suits the more light-hearted, breezy tone of the show too.